1. Happiness comes not from material wealth but less desire.

2. Statistical Package Usage
Topic: Tests for two samples
By Prof Kelly Fan, Cal State Univ, East Bay

3. Example: Mortar Strength
Unmodified modified

4. SAS Output Ho: The mean strength of the two formulations are the same
The TTEST Procedure
Statistics
Lower CL Upper CL Lower CL Upper CL
Variable FORMULATION N Mean Mean Mean Std Dev Std Dev Std Dev Std Err
STRENGTH M
STRENGTH U
STRENGTH Diff (1-2)
Ho: The mean strength of the two formulations are the same
H1: They are different

5. Type II Error, or “ Error” Type I Error, or “ Error”
Types of Errors
H0 true
H0 false
Type II Error, or “ Error”
Good!
(Correct!)
we accept H0
Type I Error, or “ Error”
Good!
(Correct)
we reject H0

6. = Probability of Type I error = P(rej. H0|H0 true)
 = Probability of Type II error = P(acc. H0|H0 false)
We often preset . The value of  depends on the specifics of the H1 (and most often in the real world, we don’t know these specifics).

7. Suppose the Critical Value = 141:
EXAMPLE: H0 :  < 100
H1 :  >100
Suppose the Critical Value = 141: X

=100
C=141

8. These are values corresp.to a value of 25 for the Std. Dev. of X
 = P (X < 141/= 150)
= .3594
 = 150
What is ?
141
 = 150
These are values corresp.to a value of 25 for the Std. Dev. of X
 = P (X < 141/= 160)
= .2236
 = 160
141
 = 160
 = 170
 = P (X < 141/= 170)
= .1230
 = 170
141
 = P (X < 141/= 180)
 = 180
= .0594
 = P (X < 141|H0 false)
141
 = 180

9. Note:. Had  been preset at. 025 (instead of
Note: Had  been preset at .025 (instead of .05), C would have been 149 (and  would be larger); had  been preset at .10, C would have been 132 and  would be smaller.
 and  “trade off”.

10. P Value
The probability of seeing as extreme as or more extreme than what we observe, assuming Ho is true.
The smaller the p-value is, the stronger the evidence against Ho is.

11. One Population
Non-parametric
test

12. Two independent
samples
Non-parametric
test

13. Normality Tests SAS: PROC UNIVARIATE DATA=** NORMAL;
Tests for Normality
Test Statistic p Value------
Shapiro-Wilk W Pr < W
Kolmogorov-Smirnov D Pr > D >0.1500
Cramer-von Mises W-Sq Pr > W-Sq
Anderson-Darling A-Sq Pr > A-Sq
Note: For demonstration purpose, here is the result for testing
normality when treating 2 samples as a whole. But you should do the normality test for both samples separately.

14. Equal-variance Tests
SAS: PROC TTEST DATA=** ;

15. Non-parametric Tests One sample: Wilcoxon Signed-Rank Test
(Included in the output of Proc Univariate)
Two samples: Wilcoxon Rank Sum Test
(See p. 191)
Three or more samples: Kruskal-Wallis Test (Later)

16. Paired Samples
If the same set of sources are used to obtain data representing two populations, the two samples are called paired. The data might be paired:
As a result of the data from certain “before” and “after” studies
From matching two subjects to form “matched pairs”

17. Example: Repeated Measures
Each subject is measured twice:
Before treatment (control value)
After treatment (treatment value)
Subject
Control
Treatment 1 90 95 2 87 92 3
100
104 4 80 89 5
101 6
105

18. Tests for Paired Samples
Calculate the pair differences
Proceed as in one sample case
Notes:
SAS: all variables must be included in data
SPSS: create/calculate all variables we need
SAS and SPSS can also conduct paired t-tests directly